Sure! Let's go through the steps to find [tex]\( g(f(x)) \)[/tex] given the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex].
1. First, substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
We are given:
[tex]\[
f(x) = x - 7
\][/tex]
[tex]\[
g(x) = 5x + 2
\][/tex]
2. Substitute [tex]\( f(x) \)[/tex] into [tex]\( g(x) \)[/tex]:
Substitute [tex]\( x - 7 \)[/tex] for [tex]\( x \)[/tex] in [tex]\( g(x) \)[/tex]:
[tex]\[
g(f(x)) = g(x - 7)
\][/tex]
3. Evaluate [tex]\( g(x - 7) \)[/tex]:
[tex]\[
g(x - 7) = 5(x - 7) + 2
\][/tex]
Distribute the 5 inside the parentheses:
[tex]\[
g(x - 7) = 5x - 35 + 2
\][/tex]
Combine like terms:
[tex]\[
g(x - 7) = 5x - 33
\][/tex]
4. Identify the coefficients:
We see that:
[tex]\[
g(f(x)) = 5x - 33
\][/tex]
Therefore, [tex]\( g(f(x)) \)[/tex] can be written as:
[tex]\[
g(f(x)) = 5 \cdot x + (-33)
\][/tex]
So, the coefficients are 5 and -33 respectively.
Therefore:
[tex]\[
g(f(x)) = 5x - 33
\][/tex]
Hence, the coefficients in the form [tex]\([?\: , \: \square]\)[/tex] are [tex]\([5\: , \: -33]\)[/tex].
Thus,
[tex]\[
g(f(x)) = 5x - 33
\][/tex]
